Csl10 9 f15 extra

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

CONTROL SYSTEMS

B.Madhuri


Time response of a second order control system?

Second Order SystemSystem with two poles


Second order system

• Two independent energy storage elements– Exchange of energy through mechanical mass and

fluid capacitance (tank) through piston– Exchange between electrical inductance and

mechanical inertia in electric motor


Time response of a second order control system

Over-damped Time Response

a

acbb

2

42

2,1

146.1

854.7

2

459

)1(2

)9)(1(499

2

1

2

2,1

tt eKeKKtc 21321)(

General solution

2

3

1

21

21 ))((

9)(

s

K

s

K

s

K

ssssC


Time response of a second order control system?

Under-damped Time Response

a

acbb

2

42

2,1

81

81

2

322

)1(2

)9)(1(422

2

1

2

2,1

j

j

j

)cos()( d

tdAetc

General solution

nd j

j

2,1

2,1 81



Under-damped Time Response

)cos()( d

tdAetc



Un-damped Time Response

a

acbb

2

42

2,1

3

3

2

36

)1(2

)9)(1(400

2

1

2

2,1

j

j

j

)cos()( Atc

General solution



Critically-damped Time Response

a

acbb

2

42

2,1

3

2

6

)1(2

)9)(1(466

2,1

2

2,1

tt teKeKtc 11

21)(

General solution


Time response of second order control system - Summary

Over-damped

Under-damped

Un-damped

Critically-damped

Poles are real and distinct

Poles are real and imaginary

Poles are imaginary

Poles are real and repeated


Time response of second order control system - Summary


Time response of second order control system - Example

a

acbb

2

42

2,1

23.1352,2

)175(410

)1(2

)200)(1(41010

1

2,1

2

2,1

j

j

The poles are real and imaginary.Therefore UNDER-DAMPED.

)cos()( d

tdAetcGeneral solution of under-damped

)23.13cos()( 5 tAetc



2

2)(

s

sG

What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response



)6)(3(

5)(

sssG

What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response


Time response of second order control system? - Example

)20)(10(

)7(10)(

ss

ssG

What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response.


Time response of second order control system

So far, we based on the poles of the second order system

a

acbb

2

42

2,1



We are going to learn two quantities that will help us

analyze second order system

Natural Frequency

nDamping

Ratio


Time response of second order control system- Natural Frequency

Natural frequency of a second order system is the frequency of oscillation of the system without

damping

Natural Frequency n


Time response of second order control system- Natural frequency

bass

bsG

2)( Natural frequency of a

second order system is the frequency of

oscillation of the system without damping

bass

bsG

2)(

bs

bsG

2)( Poles:

njs

bjs

bs

bs

02Therefore,

2n

n

b

b


Time response of second order control system- Damping Ratio

Damping Ratio

(rad/sec)Frequency Natural

FrequencyDecay lExponentia


Time response of second order control system- Damping Ratio

Damping Ratio

n

a

2

bass

bsG

2)(



bass

bsG

2)(

22

2

2)(

nn

n

sssG

General Form



362.4

36)(

2

sssG Find damping ratio and

the natural frequency?

6

362

n

n

Natural Frequency

35.0)6(2

2.4

2.42

n

Damping Ratio



22

2

2)(

nn

n

sssG

a

acbbs

2

42

2,1

Find poles in general form.

1

2

422

22,1

22

2,1

nn

nnn

s

s


Time response of second order control system.

122,1 nns

Case 1: 0 njs 2,1



122,1 nns

Case 2: 10

22,1 1 nn js



122,1 nns

Case 3: 1 ns 2,1



122,1 nns

Case 4: 112

2,1 nns



1

1

10

0

OVERDAMPED

CRITICALLY DAMPED

UNDER DAMPED

UNDAMPED


Time response of second order control system- Example

40012

400)(

2

sssG

90090

900)(

2

sssG

22530

225)(

2

sssG

625

625)(

2 s

sG

3.0)20(2

12

20400

nUNDERDAMPED

5.1)30(2

90

30900

nOVER-DAMPED

1)15(2

30

15225

nCRITICALLY-DAMPED

0)25(2

0

25625

nUN-DAMPED


Time response of second order control system. Special Case: Performance of under-damped System

Under-damped:

10 Four performance parameters:1.Rise time2.Peak time3.Percent overshoot4.Settling time



Rise TimeGeneral

definition: The time required for the waveform to go from 0.1 of the final value to 0.9 of the final value

rnT

Underdamped case: The time taken to reach the final value the first time.



Peak Time

The time required to reach the first maximum peak

21

n

pT



Percent Overshoot

The amount that the waveform overshoots the steady-state

21

100%

eOS



Settling Time

The time required for the transient damped oscillations to reach and stay within 2% of the steady-state value

nsT

4


Time response of second order control system. Special Case: Performance of under-damped System – Example

36116

361)(

2

sssG Find: OSTTT rpsn ,%,,,,

19361 n

421.0)19(2

16

5.0)19(421.0

44

nsT

182.0421.01191 22

n

pT

%3.23100100%22

421.01

421.0

1

eeOS


Time response of second order control system. Special Case: Performance of under-damped System – Example

36116

361)(

2

sssG

Find:rT

19n

421.0

079.019

501.1

501.1

r

nr

T

T

Or interpolate


Time response of second order control system? –Lessons

April 14, 2023 37

11

10

0

OVERDAMPED

CRITICALLY DAMPED

UNDER DAMPED

UNDAMPED

WHERE DO WE GET

22

2

2)(

nn

n

sssG


Time response of second order control system? -

April 14, 2023 38

What Interpretations?

Performances of second order under-damped control system are determined by two quantities called natural frequency and damping ratioFour performance parameters:1.Rise time2.Peak time3.Percent overshoot4.Settling time

21

n

pT

nsT

421100%

eOS

nr

fT

)(



April 14, 2023 39

a

acbb

2

42

2,1

22

2

2)(

nn

n

sssG

22,1 1 nn js

dd js 2,1

For under-damped system

)cos(



April 14, 2023 40

nsT

4

Line of constant settling time


Time response of second order control system?

April 14, 2023 41

21

n

pT

Line of constant peak time



April 14, 2023 42

21100%

eOS)cos(

Line of constant %OS



April 14, 2023 43

Constant Settling Time



April 14, 2023 44

Constant Peak Time



April 14, 2023 45

Constant Percent Overshoot

Engineering

Csl10 9 f15 extra